Theorem ltexprlemloc 7439

Description: Our constructed difference is located. Lemma for ltexpri 7445. (Contributed by Jim Kingdon, 17-Dec-2019.)

Hypothesis

Ref	Expression
ltexprlem.1	|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.

Assertion

Ref	Expression
ltexprlemloc	|- ( A <P B -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )

Distinct variable groups:   x, y, q, r, A   x, B, y, q, r   x, C, y, q, r

Proof of Theorem ltexprlemloc

Dummy variables z w f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexnqi 7241	. . . . . 6 |- ( q <Q r -> E. w e. Q. ( q +Q w ) = r )
2	1	adantl 275	. . . . 5 |- ( ( A <P B /\ q <Q r ) -> E. w e. Q. ( q +Q w ) = r )
3		ltrelpr 7337	. . . . . . . . . 10 |- <P C_ ( P. X. P. )
4	3	brel 4599	. . . . . . . . 9 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
5	4	simpld 111	. . . . . . . 8 |- ( A <P B -> A e. P. )
6		prop 7307	. . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
7		prarloc 7335	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ w e. Q. ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
8	6, 7	sylan 281	. . . . . . . 8 |- ( ( A e. P. /\ w e. Q. ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
9	5, 8	sylan 281	. . . . . . 7 |- ( ( A <P B /\ w e. Q. ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
10	9	ad2ant2r 501	. . . . . 6 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
11	4	simprd 113	. . . . . . . . . . . . . 14 |- ( A <P B -> B e. P. )
12	11	ad2antrr 480	. . . . . . . . . . . . 13 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> B e. P. )
13	12	ad2antrr 480	. . . . . . . . . . . 12 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ y <Q ( z +Q w ) ) -> B e. P. )
14		ltanqg 7232	. . . . . . . . . . . . . . . 16 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
15	14	adantl 275	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
16		elprnqu 7314	. . . . . . . . . . . . . . . . . . 19 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
17	6, 16	sylan 281	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
18	5, 17	sylan 281	. . . . . . . . . . . . . . . . 17 |- ( ( A <P B /\ y e. ( 2nd ` A ) ) -> y e. Q. )
19	18	adantlr 469	. . . . . . . . . . . . . . . 16 |- ( ( ( A <P B /\ q <Q r ) /\ y e. ( 2nd ` A ) ) -> y e. Q. )
20	19	ad2ant2rl 503	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> y e. Q. )
21		elprnql 7313	. . . . . . . . . . . . . . . . . . . 20 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
22	6, 21	sylan 281	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
23	5, 22	sylan 281	. . . . . . . . . . . . . . . . . 18 |- ( ( A <P B /\ z e. ( 1st ` A ) ) -> z e. Q. )
24	23	adantlr 469	. . . . . . . . . . . . . . . . 17 |- ( ( ( A <P B /\ q <Q r ) /\ z e. ( 1st ` A ) ) -> z e. Q. )
25	24	ad2ant2r 501	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> z e. Q. )
26		simplrl 525	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> w e. Q. )
27		addclnq 7207	. . . . . . . . . . . . . . . 16 |- ( ( z e. Q. /\ w e. Q. ) -> ( z +Q w ) e. Q. )
28	25, 26, 27	syl2anc 409	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( z +Q w ) e. Q. )
29		ltrelnq 7197	. . . . . . . . . . . . . . . . . . 19 |- <Q C_ ( Q. X. Q. )
30	29	brel 4599	. . . . . . . . . . . . . . . . . 18 |- ( q <Q r -> ( q e. Q. /\ r e. Q. ) )
31	30	simpld 111	. . . . . . . . . . . . . . . . 17 |- ( q <Q r -> q e. Q. )
32	31	adantl 275	. . . . . . . . . . . . . . . 16 |- ( ( A <P B /\ q <Q r ) -> q e. Q. )
33	32	ad2antrr 480	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> q e. Q. )
34		addcomnqg 7213	. . . . . . . . . . . . . . . 16 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
35	34	adantl 275	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
36	15, 20, 28, 33, 35	caovord2d 5948	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( y <Q ( z +Q w ) <-> ( y +Q q ) <Q ( ( z +Q w ) +Q q ) ) )
37		addassnqg 7214	. . . . . . . . . . . . . . . . 17 |- ( ( z e. Q. /\ w e. Q. /\ q e. Q. ) -> ( ( z +Q w ) +Q q ) = ( z +Q ( w +Q q ) ) )
38	25, 26, 33, 37	syl3anc 1217	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( ( z +Q w ) +Q q ) = ( z +Q ( w +Q q ) ) )
39		addcomnqg 7213	. . . . . . . . . . . . . . . . . 18 |- ( ( w e. Q. /\ q e. Q. ) -> ( w +Q q ) = ( q +Q w ) )
40	26, 33, 39	syl2anc 409	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( w +Q q ) = ( q +Q w ) )
41	40	oveq2d 5798	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( z +Q ( w +Q q ) ) = ( z +Q ( q +Q w ) ) )
42		simplrr 526	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( q +Q w ) = r )
43	42	oveq2d 5798	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( z +Q ( q +Q w ) ) = ( z +Q r ) )
44	38, 41, 43	3eqtrd 2177	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( ( z +Q w ) +Q q ) = ( z +Q r ) )
45	44	breq2d 3949	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( ( y +Q q ) <Q ( ( z +Q w ) +Q q ) <-> ( y +Q q ) <Q ( z +Q r ) ) )
46	36, 45	bitrd 187	. . . . . . . . . . . . 13 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( y <Q ( z +Q w ) <-> ( y +Q q ) <Q ( z +Q r ) ) )
47	46	biimpa 294	. . . . . . . . . . . 12 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ y <Q ( z +Q w ) ) -> ( y +Q q ) <Q ( z +Q r ) )
48		prop 7307	. . . . . . . . . . . . 13 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
49		prloc 7323	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ ( y +Q q ) <Q ( z +Q r ) ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) )
50	48, 49	sylan 281	. . . . . . . . . . . 12 |- ( ( B e. P. /\ ( y +Q q ) <Q ( z +Q r ) ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) )
51	13, 47, 50	syl2anc 409	. . . . . . . . . . 11 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ y <Q ( z +Q w ) ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) )
52	51	ex 114	. . . . . . . . . 10 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( y <Q ( z +Q w ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
53	52	anassrs 398	. . . . . . . . 9 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ z e. ( 1st ` A ) ) /\ y e. ( 2nd ` A ) ) -> ( y <Q ( z +Q w ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
54	53	reximdva 2537	. . . . . . . 8 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ z e. ( 1st ` A ) ) -> ( E. y e. ( 2nd ` A ) y <Q ( z +Q w ) -> E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
55	54	reximdva 2537	. . . . . . 7 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
56		prml 7309	. . . . . . . . . . . 12 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. z e. Q. z e. ( 1st ` A ) )
57		rexex 2482	. . . . . . . . . . . 12 |- ( E. z e. Q. z e. ( 1st ` A ) -> E. z z e. ( 1st ` A ) )
58	6, 56, 57	3syl 17	. . . . . . . . . . 11 |- ( A e. P. -> E. z z e. ( 1st ` A ) )
59		r19.45mv 3461	. . . . . . . . . . 11 |- ( E. z z e. ( 1st ` A ) -> ( E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
60	5, 58, 59	3syl 17	. . . . . . . . . 10 |- ( A <P B -> ( E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
61	60	adantr 274	. . . . . . . . 9 |- ( ( A <P B /\ q <Q r ) -> ( E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
62		prmu 7310	. . . . . . . . . . . . 13 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. x e. Q. x e. ( 2nd ` A ) )
63		rexex 2482	. . . . . . . . . . . . 13 |- ( E. x e. Q. x e. ( 2nd ` A ) -> E. x x e. ( 2nd ` A ) )
64	6, 62, 63	3syl 17	. . . . . . . . . . . 12 |- ( A e. P. -> E. x x e. ( 2nd ` A ) )
65		r19.43 2592	. . . . . . . . . . . . 13 |- ( E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. y e. ( 2nd ` A ) ( z +Q r ) e. ( 2nd ` B ) ) )
66		r19.9rmv 3459	. . . . . . . . . . . . . 14 |- ( E. x x e. ( 2nd ` A ) -> ( ( z +Q r ) e. ( 2nd ` B ) <-> E. y e. ( 2nd ` A ) ( z +Q r ) e. ( 2nd ` B ) ) )
67	66	orbi2d 780	. . . . . . . . . . . . 13 |- ( E. x x e. ( 2nd ` A ) -> ( ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. y e. ( 2nd ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
68	65, 67	bitr4id 198	. . . . . . . . . . . 12 |- ( E. x x e. ( 2nd ` A ) -> ( E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
69	5, 64, 68	3syl 17	. . . . . . . . . . 11 |- ( A <P B -> ( E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
70	69	rexbidv 2439	. . . . . . . . . 10 |- ( A <P B -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
71	70	adantr 274	. . . . . . . . 9 |- ( ( A <P B /\ q <Q r ) -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
72		ltexprlem.1	. . . . . . . . . . . . . 14 |- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.
73	72	ltexprlemell 7430	. . . . . . . . . . . . 13 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
74	72	ltexprlemelu 7431	. . . . . . . . . . . . . 14 |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
75		eleq1 2203	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( y e. ( 1st ` A ) <-> z e. ( 1st ` A ) ) )
76		oveq1 5789	. . . . . . . . . . . . . . . . . 18 |- ( y = z -> ( y +Q r ) = ( z +Q r ) )
77	76	eleq1d 2209	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( ( y +Q r ) e. ( 2nd ` B ) <-> ( z +Q r ) e. ( 2nd ` B ) ) )
78	75, 77	anbi12d 465	. . . . . . . . . . . . . . . 16 |- ( y = z -> ( ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) <-> ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
79	78	cbvexv 1891	. . . . . . . . . . . . . . 15 |- ( E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) <-> E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) )
80	79	anbi2i 453	. . . . . . . . . . . . . 14 |- ( ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
81	74, 80	bitri 183	. . . . . . . . . . . . 13 |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
82	73, 81	orbi12i 754	. . . . . . . . . . . 12 |- ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) \/ ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
83		ibar 299	. . . . . . . . . . . . . . 15 |- ( q e. Q. -> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) )
84	83	adantr 274	. . . . . . . . . . . . . 14 |- ( ( q e. Q. /\ r e. Q. ) -> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) )
85		ibar 299	. . . . . . . . . . . . . . 15 |- ( r e. Q. -> ( E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
86	85	adantl 275	. . . . . . . . . . . . . 14 |- ( ( q e. Q. /\ r e. Q. ) -> ( E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
87	84, 86	orbi12d 783	. . . . . . . . . . . . 13 |- ( ( q e. Q. /\ r e. Q. ) -> ( ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) <-> ( ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) \/ ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) ) )
88	30, 87	syl 14	. . . . . . . . . . . 12 |- ( q <Q r -> ( ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) <-> ( ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) \/ ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) ) )
89	82, 88	bitr4id 198	. . . . . . . . . . 11 |- ( q <Q r -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
90		df-rex 2423	. . . . . . . . . . . 12 |- ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) <-> E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) )
91		df-rex 2423	. . . . . . . . . . . 12 |- ( E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) <-> E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) )
92	90, 91	orbi12i 754	. . . . . . . . . . 11 |- ( ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
93	89, 92	syl6bbr 197	. . . . . . . . . 10 |- ( q <Q r -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
94	93	adantl 275	. . . . . . . . 9 |- ( ( A <P B /\ q <Q r ) -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
95	61, 71, 94	3bitr4rd 220	. . . . . . . 8 |- ( ( A <P B /\ q <Q r ) -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
96	95	adantr 274	. . . . . . 7 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
97	55, 96	sylibrd 168	. . . . . 6 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
98	10, 97	mpd 13	. . . . 5 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) )
99	2, 98	rexlimddv 2557	. . . 4 |- ( ( A <P B /\ q <Q r ) -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) )
100	99	ex 114	. . 3 |- ( A <P B -> ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
101	100	ralrimivw 2509	. 2 |- ( A <P B -> A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
102	101	ralrimivw 2509	1 |- ( A <P B -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   /\ w3a 963   = wceq 1332   E.wex 1469   e. wcel 1481   A.wral 2417   E.wrex 2418   {crab 2421   <.cop 3535   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   1stc1st 6044   2ndc2nd 6045   Q.cnq 7112   +Q cplq 7114   <Q cltq 7117   P.cnp 7123   <P cltp 7127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-iltp 7302

This theorem is referenced by:  ltexprlempr  7440